Lemma D.6 (Lemma D.4 in [1]}) 
Let \(\lbrace s_i\rbrace _{i=1}^\infty \)  be a stochastic process on state space \(\)  with corresponding filtration \(\lbrace _i\rbrace _{i=1}^\infty \) . Let \(\lbrace \phi _i\rbrace _{i=1}^\infty \)  be an \(\mathbb {R}^d\) -valued stochastic process where \(\phi _i \in _{i-1}\) , and \(\Vert \phi _i\Vert  \le 1\) . Let \(\Lambda _k = \lambda I + \sum _{i=1}^k \phi _i \phi _i^\top \) . Then for any \(\delta > 0\) , with probability at least \(1-\delta \) , for all \(k \ge 0\) , and any \(V \in \)  with \(\sup _{s \in } |V(s)| \le H\) , we have
\(\Bigl \Vert \sum _{i=1}^k \phi _i \bigl \lbrace  V(s_i) - \mathbb {E}[V(s_i) {\,|\,}_{i-1} ]\bigr \rbrace  \Bigr \Vert ^2_{\Lambda _k^{-1}} \le 4H^2 \Bigl [\frac{d}{2}\log \Bigl (\frac{k+\lambda }{\lambda }\Bigr ) + \log {\frac{_\varepsilon }{\delta }}\Bigr ] + \frac{8k^2\epsilon ^2}{\lambda },\) 
